The White House declined to disclose what decision Mr. Obama had reached, but some administration officials believe it is unlikely that he will resolve the contentious issue in the waning moments of his presidency.
That such a pivotal decision has been left to Mr. Obama’s final weeks in office reflects the complexity of the debate about working with the Y.P.G., as the Syrian Kurdish militia is known, as well as the caution the president has displayed about sending American forces to fight in the region.
Mr. Obama has vowed to deal the Islamic State crippling blows in Mosul, Iraq, and Raqqa before he steps down on Friday. Allied airstrikes have increased in and around Raqqa in recent weeks as thousands of Syrian Kurdish and Syrian Arab fighters encircle the city, isolating it from the resupply of arms, fighters and fuel. Last month, Mr. Obama ordered 200 more American Special Operations forces to Syria to help these local fighters advancing on Raqqa, nearly doubling the number of American troops on the ground there.
But the American military believes that Raqqa cannot be seized unless the Y.P.G. is equipped for urban warfare. It is unclear what level of support President-elect Donald J. Trump will maintain for opposition groups in Syria combating the Islamic State, especially those groups that are bitterly opposed by the Turks.
Defense Secretary Ashton B. Carter stressed Raqqa’s importance during a visit to Fort Campbell, Ky., in January 2016. “The ISIL parent tumor has two centers: Raqqa in Syria and Mosul in Iraq,” Mr. Carter said. “That’s why our campaign plan’s got big arrows pointing at both Mosul and Raqqa.”
American officials requested anonymity in order to describe the administration’s internal deliberations.
About 250,000 civilians are in Raqqa, and the Islamic State has fortified the city with trenches and mines and would defend it with suicide bombers. Because the Obama administration has ruled out the use of American combat troops, the United States has to rely on mobilizing local Arab forces to join battle-hardened Syrian Kurdish fighters.
“Raqqa is very difficult because unlike Iraq, we’re not working with a government,” Brett McGurk, the American envoy to the coalition that is fighting the Islamic State, said at a seminar last week. “We’re not working with an army. We have to work with local actors and organize them into a military force.”
American military officials say it is urgent to retake Raqqa because it is the capital of the Islamic State’s caliphate, a sanctuary for many of its top leaders and the hub for the extremist group’s plots against the West.
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Tyler Hicks/The New York Times
The Pentagon has been urging Mr. Obama to equip the Syrian Kurds, whom American commanders view as their most effective ground partner, with armored vehicles, rocket-propelled-grenade launchers, machine guns and other heavy equipment so that the American-supported Raqqa attack can begin in February.
The weaponry is needed, American military officials say, because the Iraqi push to capture Mosul has demonstrated that retaking a city occupied by Islamic State fighters, armed with suicide car bombs, is a difficult and bloody operation.
To buttress the Raqqa mission, the Pentagon is also urging that the White House authorize the use of United States Army Apache attack helicopters, which are equipped with Hellfire missiles. Apaches are supporting Iraqi troops in the fight for Mosul.
But arming the Kurds would also aggravate Mr. Obama’s tense relations with Turkey’s president, Recep Tayyip Erdogan, who has contended that the Y.P.G. is linked to the Kurdistan Workers’ Party, which Turkey and the United States regard as a terrorist group.
The administration has been considering ways to ease Turkey’s anxiety, such as making arrangements to monitor the weapons given to the Syrian Kurds for the Raqqa offensive and thus prevent the weapons from being used elsewhere by the Kurds. In addition, Arab forces would occupy Raqqa after the city is taken, and Kurdish fighters would be withdrawn.
The United States also recently began carrying out airstrikes near Al Bab, a town in northern Syria that Turkey has been struggling to take from the Islamic State.
But American diplomats in Ankara, the Turkish capital, have warned that providing weapons to the Y.P.G. could provoke a Turkish backlash, officials say. Not only might it cause a deep breach in the United States’ relations with Mr. Erdogan, but the Turks might take actions against the Y.P.G. in northern Syria that could ultimately undermine the offensive to retake Raqqa.
Anticipating Mr. Obama’s decision, the Turks have been quietly increasing the pressure by delaying approval for American air missions that are flown from the Turkish air base at Incirlik and supplies going in and out of the base. Incirlik has been a major hub for carrying out airstrikes against the Islamic State in Syria and Iraq.
Turkey’s sensitivity on the issue was clear last week when the United States Central Command, which oversees military operations in the Middle East, posted a statement on Twitter by the Syrian Democratic Forces, the umbrella group that includes Syrian Kurds as well as Syrian Arab fighters, affirming that it is not part of the Kurdistan Workers’ Party as “some regional governments” have claimed.
“Is this a joke or @CENTCOM has lost its senses,” Ibrahim Kalin, Mr. Erdogan’s spokesman, responded on Twitter.
Faced with the dilemma, some administration officials have suggested that American officials go back to the drawing board and try to cobble together a more diverse force to take Raqqa that would include Turkish Special Forces as well as Turkish-supported Syrian opposition groups. American commanders say about 20,000 troops will be needed to seize the city. By contrast, Turkey has been able to muster only about 2,000 Arab fighters in its battle to reclaim Al Bab, and that campaign has been bogged down by fierce resistance.
During a visit to Washington last month, Masrour Barzani, a top security official in the Kurdish autonomous region in Iraq, pressed American officials to work with Syrian Kurds who are separate from the Y.P.G. and are operating in Iraq, a group known as Pesh Merga of Rojava, or Roj Pesh. Aides to Mr. Barzani assert that the Roj Pesh are trained by the pesh merga, would be politically acceptable to the Turks and number about 3,300.
“Roj Pesh are the most efficient and politically diverse force,” Mr. Barzani said. “They can be the bridge to lessen regional tensions and a force multiplier in the campaign.”
But Pentagon officials say that the Y.P.G. has the most effective fighters, is already closing in on Raqqa, and that trying to assemble, train and equip an alternative force could be difficult and at best would take many months.
